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Nodal plane means that the wave function = 0, meaning that the probability of finding electrons in that plane is zero. I am not sure about why s-orbitals don't have nodal plane, but I think that it is because there is no node when depicting the math model of the s-orbital.
In the s-orbital, electorns have no angular momentum (l = 0), so they're distributed around the nucleus instead of circulating it. In p-,d-, and f- orbitals, electrons have angular momentum (l>0), and the spherical movement creates nodal planes.
S-orbitals have electrons that are distributed spherically symmetrical around the nucleus so while they do not have any nodal planes, the s-orbital does have radial nodes. The 1s orbital would not have any nodes, but the 2s orbital would have one node, the 3s orbital would have two nodes, etc.
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